The Football Association has been fined £4,500 and the Turkish authorities £13,300 over the fracas at half-time in October's Euro 2004 qualifier in Istanbul.
Uefa charged the English and Turkish football associations with improper conduct and on Thursday hit both organisations with the financial penalty.
No individual players were charged over the scuffle, which was captured by television cameras.
The game ended 0-0 - earning England a place at the Portugal finals, with Turkey facing a play-off against Latvia.
Uefa decided to pursue the associations rather than players after
considering video evidence, match reports from referee Pierluigi Collina and the match observer, as well as statements from both countries.
Turkey defender Alpay and David Beckham clashed during the first-half of the match when the England captain missed a penalty.
The duo squared up to each other again as they walked off at half-time and Collina called them into his room during the break to calm them down.
The Italian referee was also seen ordering other players to their own dressing-rooms during the flare-up.
However, Collina did not highlight any individual player as being guilty of violent behaviour in his match report.